Difference between revisions of "Sequence"

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==Convergence of a sequence==
 
==Convergence of a sequence==
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===Topological form===
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* See [[Convergence of a sequence]]
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A sequence <math>(a_n)_{n=1}^\infty</math> in a [[Topological space|topological space]] {{M|X}} converges if <math>\forall U</math> that are open neighbourhoods of {{M|x}} <math>\exists N\in\mathbb{N}: n> N\implies x_n\in U</math>
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===Metric space form===
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A sequence <math>(a_n)_{n=1}^\infty</math> in a [[Metric space|metric space]] {{M|V}} (Keep in mind it is easy to get a metric given a [[Norm|normed]] [[Vector space|vector space]]) is said to converge to a limit <math>a\in V</math> if:
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<math>\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies d(a_n,a)<\epsilon</math> - note the [[Implicit qualifier|implicit <math>\forall n</math>]]
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In this case we may write: <math>\lim_{n\rightarrow\infty}(a_n)=a</math>
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===Basic form===
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Usually <math>\forall\epsilon>0\exists N\in\mathbb{N}: n > N\implies |a_n-a|<\epsilon</math> is first seen, or even just a [[Null sequence]] then defining converging to {{M|a}} by subtraction, like with [[Continuous map]] you move on to a metric space.
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===Normed form===
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In a [[Norm|normed]] [[Vector space|vector space]] as you'd expect it's defined as follows:
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<math>\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies\|a_n-a\|<\epsilon</math>, note this it the definition of the sequence <math>(\|a_n-a\|)_{n=1}^\infty</math> tending towards 0
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==See also==
 
==See also==
 
* [[Cauchy criterion for convergence]]
 
* [[Cauchy criterion for convergence]]
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* [[Convergence of a sequence]]
  
 
==References==
 
==References==
  
 
{{Definition|Set Theory|Real Analysis|Functional Analysis}}
 
{{Definition|Set Theory|Real Analysis|Functional Analysis}}

Revision as of 17:13, 8 March 2015

Introduction

A sequence is one of the earliest and easiest definitions encountered, but I will restate it.

I was taught to denote the sequence [math]\{a_1,a_2,...\}[/math] by [math]\{a_n\}_{n=1}^\infty[/math] however I don't like this, as it looks like a set. I have seen the notation [math](a_n)_{n=1}^\infty[/math] and I must say I prefer it.

Definition

Formally a sequence is a function[1], [math]f:\mathbb{N}\rightarrow S[/math] where [ilmath]S[/ilmath] is some set. For a finite sequence it is simply [math]f:\{1,...,n\}\rightarrow S[/math]

There is little more to say.

Convergence of a sequence

See also

References

  1. ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
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